The dynamical equivalence of modified gravity revisited

We revisit the dynamical equivalence between different representations of vacuum modified gravity models in view of Legendre transformations. The equivalence is discussed for both bulk and boundary space, by including in our analysis the relevant Gibbons-Hawking terms. In the f(R) case, the Legendre transformed action coincides with the usual Einstein frame one. We then re-express the R+f(G) action, where G is the Gauss-Bonnet term, as a second order theory with a new set of field variables, four tensor fields and one scalar and study its dynamics. For completeness, we also calculate the conformal transformation of the full Jordan frame R+f(G) action. All the appropriate Gibbons-Hawking terms are calculated explicitly.